The general real function-algebra definition agrees with the published compact-metric definition

Given: A nonempty compact metric space $(K,d)$ with its metric topology, and a subset $A$ of its real-valued continuous functions.

For nonempty compact metric $K$, a subset of $C(K,\mathbb{R})$ is a unital real function algebra when it contains every constant function and is closed under pointwise addition, real scalar multiplication, and multiplication; it separates points when every distinct pair is distinguished by one member (A unital point-separating real subalgebra of $C(K,\mathbb{R})$).

The metric-space notation $C(K,\mathbb{R})$ consists of the continuous functions from $(K,d)$ to $\mathbb{R}$ with its usual metric (The space $C(K,\mathbb{R})$ of continuous real-valued functions on a nonempty compact metric space).

For maps between metric spaces, epsilon-delta continuity at every point is equivalent to the inverse image of every open set being open (Metric continuity characterisations, with countable choice for the sequential converse).

A metric space is compact if and only if it is compact as a topological space in its metric topology (For a metric space with its metric topology, compactness in the topological sense is compactness in the metric sense, and the two notions of compact subset coincide).

Every metric space is Hausdorff: distinct points are separated by disjoint open balls (Distinct points of a metric space have disjoint balls around them).

A real function algebra on a compact Hausdorff space is a real vector subspace of $C(K,\mathbb{R})$ closed under pointwise multiplication; unitality means that it contains every constant function, and point separation means that every distinct pair is distinguished by one member (Unital, point-separating, and nowhere-vanishing real function algebras on a compact Hausdorff space).